Path integral action for a resonant detector of gravitational waves in the generalized uncertainty principle framework

The Heisenberg uncertainty principle gets modified by the introduction of an observer independent minimal length. In this work we have considered the resonant gravitational wave detector in the modified uncertainty principle framework where we have used the position momentum uncertainty relation with a quadratic order correction only. We have then used the path integral approach to calculate an action for the bar detector in presence of a gravitational wave and then derived the Lagrangian of the system leading to the equation of motion for the configuration-space position coordinate in one dimension. We then find a perturbative solution for the coordinate of the detector for a circularly polarized gravitational wave leading to a classical solution of the same for given initial conditions. Using this classical form of the coordinate of the detector, we finally obtain the classical form of the on-shell action describing the harmonic oscillator-gravitational wave system. Finally, we have obtained the free particle propagator containing the quantum fluctuation term considering gravitational wave interaction.